Algebra and Sequences

Algebra and sequences appear regularly on the CSSE 11+ maths paper. The CSSE is known for pushing students beyond simple arithmetic — you may be asked to find unknown values, write expressions, and spot patterns in number sequences. This lesson covers all the algebra and sequence skills you need.

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What Is Algebra?

Algebra uses letters to stand for unknown numbers. Instead of writing "a number", we write a letter such as n, x, or a.

Expression	Meaning
3n	3 × n (three times n)
n + 7	n plus 7
2n - 4	Two times n, minus 4
n/5	n divided by 5
n²	n × n (n squared)

Remember: In algebra, we do not usually write the multiplication sign. We write 3n instead of 3 × n.

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Simplifying Expressions

To simplify, collect like terms — terms that contain the same letter.

Simplify: 4a + 3b + 2a - b

• Collect the a terms: 4a + 2a = 6a
• Collect the b terms: 3b - b = 2b
• Answer: 6a + 2b

Simplify: 5x + 3 - 2x + 7

• Collect x terms: 5x - 2x = 3x
• Collect number terms: 3 + 7 = 10
• Answer: 3x + 10

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Substitution

Substitution means replacing a letter with a number and calculating the result.

If a = 5 and b = 3, find the value of 2a + 4b - 7

• 2 × 5 + 4 × 3 - 7
• = 10 + 12 - 7
• = 15

If x = 4, find the value of x² + 3x

• 4² + 3 × 4
• = 16 + 12
• = 28

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Solving Equations

An equation has an equals sign. Solving it means finding the value of the unknown letter.

One-Step Equations

Solve: x + 8 = 15

• x = 15 - 8 = 7

Solve: 3y = 21

• y = 21 ÷ 3 = 7

Two-Step Equations

Solve: 2x + 5 = 19

Step 1: Subtract 5 from both sides: 2x = 14

Step 2: Divide both sides by 2: x = 7

Solve: 3n - 4 = 17

Step 1: Add 4 to both sides: 3n = 21

Step 2: Divide both sides by 3: n = 7

CSSE Tip: Equation questions on the CSSE are often disguised as word problems. For example: "I think of a number, double it, and add 5. The answer is 23. What is my number?"

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Forming Equations from Word Problems

I think of a number, multiply it by 4, and subtract 3. The answer is 25. What is the number?

• Let the number be n
• 4n - 3 = 25
• 4n = 28
• n = 7

The perimeter of a rectangle is 38 cm. The width is 7 cm. What is the length?

• Let the length be l
• 2l + 2 × 7 = 38
• 2l + 14 = 38
• 2l = 24
• l = 12 cm

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Function Machines

A function machine takes an input, applies one or more operations, and produces an output.

Input --> [× 4] --> [- 3] --> Output

If the input is 6:

• 6 × 4 = 24
• 24 - 3 = 21

Working backwards: If the output is 25, what is the input?